Space-Efficient Algorithms for Reachability in Geometric Graphs
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The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem. For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and O(m^1/2log n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial-time and O(m^1/2log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, we use a more involved technique for unit contact disk graphs (penny graphs) and obtain a better algorithm. We show that for every ϵ> 0, there exists a polynomial-time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n^1/4+ϵ) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.
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